an improved virtual inertia control for three-phase

Aalborg Universitet

An Improved Virtual Inertia Control for Three-Phase Voltage Source ConvertersConnected to a Weak Grid

Fang, Jingyang; Lin, Pengfeng; Li, Hongchang; Yang, Yongheng; Tang, Yi

Published in:I E E E Transactions on Power Electronics

DOI (link to publication from Publisher):10.1109/TPEL.2018.2885513

Publication date:2019

Document VersionAccepted author manuscript, peer reviewed version

Link to publication from Aalborg University

Citation for published version (APA):Fang, J., Lin, P., Li, H., Yang, Y., & Tang, Y. (2019). An Improved Virtual Inertia Control for Three-PhaseVoltage Source Converters Connected to a Weak Grid. I E E E Transactions on Power Electronics, 34(9), 8660-8670. [8567928]. https://doi.org/10.1109/TPEL.2018.2885513

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2018.2885513, IEEETransactions on Power Electronics

An Improved Virtual Inertia Control for Three-

Phase Voltage Source Converters Connected to a

Weak Grid

Abstract—The still increasing share of power converter-

based renewable energies weakens the power system inertia.

The lack of inertia becomes a main challenge to small-scale

modern power systems in terms of control and stability. To

alleviate such adverse effects from inertia reductions, e.g.,

undesirable load shedding and cascading failures, three-phase

grid-connected power converters should provide virtual inertia

upon system demands. This can be achieved through directly

linking the grid frequency and voltage references of DC-link

capacitors/ultracapacitors. This paper reveals that the virtual

inertia control may possibly induce instabilities to the power

converters under weak grid conditions, which is caused by the

coupling between the d- and q-axis as well as the inherent

differential operator introduced by the virtual inertia control.

To tackle this instability issue, this paper proposes a modified

virtual inertia control to mitigate the differential effect, and thus

alleviating the coupling effect to a great extent. Experimental

verifications are provided, which demonstrate the effectiveness

of the proposed control in stabilizing three-phase grid-

connected power converters for inertia emulation even when

connected to weak grids.

Index Terms—Frequency regulation, renewable energy,

stability, virtual inertia, weak grid, power converter.

I. INTRODUCTION

HE main driving forces to deploy and develop a large

amount of renewable energies integrated into modern

power systems include the reduction of carbon footprint and

increment of clean energy production [1]. Despite being

pursued worldwide, the employment of renewable energies

even partly to replace fossil fuels (expected to completely

phase out in the future) may retrofit the entire power grid and

challenge the stability of modern power systems [2]. One of

the major concerns, which has already been acknowledged in

small-scale power systems, e.g., in Ireland and Great Britain,

refers to the frequency instability due to the lack of inertia in

the system with a high penetration level of power converter-

interfaced renewable energies [3].

In conventional power systems, synchronous generators

operating at a speed in synchronism with the grid frequency

act as the major grid interfaces [4]. When a frequency change

due to the power imbalance between generation and demand

occurs, synchronous generators autonomously slow down or

speed up in accordance with the grid frequency. In this way,

the synchronous generators release/absorb the energy to/from

the power grid so that the power mismatch can partially be

compensated. This effect is quantitatively evaluated by the

per unit kinetic energy, which is defined as power system

inertia [4], [5]. However, this phenomenon changes in

modern power systems, because most of renewable energy

sources (RESs), such as wind energies and solar

photovoltaics (PV), are coupled to the power grid through

power electronic converters [1], [6]. Different from

synchronous generators, grid-connected power converters

normally operate in the maximum power point tracking

(MPPT) mode to optimize the energy yield without any

inertia contribution [7]. This is because there is no kinetic

energy in such power conversion systems that can be used as

rotational inertia [8]. As more synchronous generators are

phasing out and being replaced by power converters, the

entire power system becomes more inertia-less, being a major

concern for stability and control. More specifically, without

sufficient inertia, the grid frequency and/or the rate-of-

change-of-frequency (RoCoF) may be apt to go beyond the

acceptable range under severe frequency events, leading to

generation tripping, undesirable load shedding, or even

system collapses [9].

Various solutions to tackle the lack of inertia issue have

already been proposed in the literature. One straightforward

approach is to change the requirements of RoCoF withstand

capabilities of generators [3]. Although this solution has been

accepted as an efficient and suitable solution by the system

operators in Ireland/North Ireland [10], the high costs

associated with generator testing mainly hinder its wide

applications. In addition, since this solution involves no effort

Jingyang Fang, Student Member, IEEE, Pengfeng Lin, Student Member, IEEE, Hongchang Li,

Member, IEEE, Yongheng Yang, Senior Member, IEEE, Yi Tang, Senior Member, IEEE

Manuscript received June 06, 2018; revised September 03, 2018 and

October 31, 2018; accepted December 04, 2018. This research is supported

by the National Research Foundation, Prime Minister’s Office, Singapore under the Energy Innovation Research Programme (EIRP) Energy Storage

Grant Call and administrated by the Energy Market Authority

(NRF2015EWT-EIRP002-007). The paper has been presented in part in [43]. (Corresponding author: Yi Tang)

J. Fang and Y. Tang are with the School of Electrical and Electronic

Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]; [email protected]).

H. Li and P. Lin are with the Energy Research Institute @ NTU

(ERI@N), Nanyang Technological University, Singapore 639798 (e-mail: [email protected]; [email protected];).

Y. Yang is with the Department of Energy Technology, Aalborg

University, Aalborg 9220, Denmark (e-mail: [email protected]).

T



for inertia enhancement, the inertia issue cannot be

completely resolved. Another possibility for inertia

enhancement is the use of synchronous condensers, i.e.,

synchronous generators without prime movers or loads [11].

However, high capital and operating costs have deterred the

widespread adoption of synchronous condensers.

Similar to synchronous generators, wind turbines also

feature rotating masses and the associated kinetic energy.

However, their rotating speeds are usually decoupled from

the grid frequency for a better speed control to optimize the

energy harvesting [12], [13]. In this regard, variable-speed

wind generation systems normally contribute zero

synchronous inertia to the power grid. To exploit the stored

kinetic energy in wind turbines, the electromagnetic torque

(or grid-injected power) can be changed in response to the

grid frequency during frequency events [13]. Through this

approach, the emulated inertia or virtual inertia will be

synthesized by wind turbines. In [14], the electromagnetic

torque is proportionally linked to the RoCoF (i.e., df / dt) for

inertia emulation. References [15−17] propose several

modified virtual inertia controls without considering the

aerodynamics and speed recovery processes of wind turbines.

As analyzed by [18] and [19], speed recovery processes will

greatly change the inertia response of wind turbines and may

cause a recurring frequency dip or even rotor stall. As such,

the emulated inertia from wind turbines and synchronous

inertia are still not identical, as proven by the wind inertial

response in Hydro-Quebec [3].

Another emerging approach aims to generate virtual

inertia through battery storage systems. This can be achieved

by introducing a proportional link between RoCoF (i.e., df /

dt) and battery power references [20−22]. As mentioned,

similar inertia emulation schemes have already been applied

to wind turbines [23], [24]. The formulation of battery power

references necessitates the fast and accurate detection of

RoCoF signals, which is considered as a significant challenge

by the power system operators in Ireland and Great Britain

[3]. To tackle this issue, the possibility of using a frequency-

locked-loop (FLL) for RoCoF detection is discussed in [21].

In addition, it is revealed that power converters may have

instability concerns when adopting the df / dt-based inertia

emulation scheme [25], [26]. Furthermore, slow inertia

emulation (which can be achieved through the use of a first-

order lag of a time constant around 1 s) can address the

instability concerns [26]. The analysis and conclusion

provided in [26] are of great importance and can provide

useful guidelines for virtual inertia design.

As compared to batteries, ultracapacitors feature a higher

power density and are therefore suitable for inertia emulation

[27]. In addition to ultracapacitors, capacitors are normally

necessary in the DC-links of grid-connected power

converters for voltage support and harmonic filtering [28].

These capacitors may also be exploited to reap the benefit of

their capabilities for inertia emulation with a small or even no

change of hardware [29], [30]. Regarding control

implementations, the inertia emulated by capacitors and

ultracapacitors are very similar. Because of adjustable

capacitor and ultracapacitor voltages, RoCoF detection is no

longer necessary. By proportionally linking the capacitor

voltage and the grid frequency, virtual inertia can be expected

from the capacitors/ultracapacitors [29], [31], [32]. This

approach can be very effective and simple in terms of control

designs. Therefore, it has been regarded as a promising

solution and extended to high-voltage direct current (HVDC)

applications such as modular multilevel converters (MMCs)

[33−35].

It is known that weak grids featuring large grid

impedances may make three-phase grid-connected power

converters unstable due to the resonance caused by high-

order passive filters [36], coupling among multiple power

converters [37], and influence of the phase-locked-loop

(PLL) on the current control [38]. In addition to the above-

mentioned causes, this paper reveals that the strong coupling

between the control loops (i.e., the dq-reference control

systems) and the differential operator are responsible for the

instability of grid-connected power converters with virtual

inertia from capacitors/ultracapacitors.

The rest of this paper is organized as follows. Section II

presents the concept of virtual inertia and the fundamentals

of three-phase power converters with the virtual inertia

control. With the system models, the instability issue is

further explored in Section III. Section IV introduces the

proposed modified virtual inertia control for stability

enhancement. The effectiveness of the proposed control to

enhance the stability under weak grids is experimentally

verified in Section V. Finally, Section VI provides the

concluding remarks.

II. THREE-PHASE GRID-CONNECTED POWER

CONVERTERS WITH VIRTUAL INERTIA CONTROL

Power electronics is the key to renewable energy

integration, and power converters as grid interfaces are

replacing conventional synchronous generators. However,

presently, the power system inertia is solely contributed by

the rotating masses of synchronous generators. As a result,

the lack of inertia may challenge the operation and control of

modern power systems [3], [9].

A. Concept of Virtual Inertia

To address the issue of less inertia in more-electronics

power systems, capacitors and/or ultracapacitors have been

increasingly exploited to absorb/release power in a similar

way as conventional synchronous generators do in the case of

frequency events for inertia emulation [29−32]. Fig. 1

illustrates the mapping between the synchronous generator

and the capacitor, where the inertia constant of the

synchronous generator H is defined as the ratio of the kinetic



energy stored in the rotating masses of the synchronous

generator (Jω0m2 / 2) to its rated power Sbase, which can be

expressed as [4] 2

0

base

,2

mJH

S

= (1)

where J denotes the combined moment of inertia of the

synchronous generator and turbine. ω0m stands for the rated

rotor mechanical speed. Similarly, the inertia constant of the

capacitor Hcap can be defined as the ratio of the electrical

energy stored in the capacitor (CdcVdc_ref2 / 2) to its rated

power Sbase, which can be expressed as [29] 2

dc dc_ref

cap

base

,2

C VH

S= (2)

in which Cdc and Vdc_ref represent the capacitance and rated

capacitor voltage, respectively. The rotor speed ωr (note that

the electrical speed ωr equals the mechanical speed ωm for

synchronous generators with one pair of poles) and capacitor

voltage vdc have the same role in determining the

corresponding inertia constant, H and Hcap, as noticed in (1)

and (2). Therefore, the virtual inertia with an equivalent

inertia coefficient of HcapKωv_pu can be expected from the

capacitor after the per unit capacitor voltage and the per unit

rotor speed or the grid frequency are linked through the

proportional gain Kωv_pu, where [29]

dc_pu _pu _pu .v rv K = (3)

B. Power Converters with Virtual Inertia Control

Fig. 2 shows the schematic diagram of a three-phase grid-

connected power converter with the virtual inertia control,

where the power grid is modelled as a series connection of an

ideal voltage source vabc and a grid inductor Lg. In the case of

weak grids (large Lg), the voltage drop across Lg can be

considerable, thereby making the measured voltages at the

point of common coupling (PCC) vgabc significantly different

from the ideal grid voltages vabc. A two-level three-phase

power converter with an output inductor filter Lc is employed

to investigate the potential instability issue due to inertia

emulation. As shown in Fig. 2, the control structure of the

system consists of two parts – a virtual inertia controller and

a conventional cascaded voltage/current controller

implemented in the synchronous dq-frame [29], [30]. The

virtual inertia controller is expected to change the DC-link

voltage reference by providing a voltage difference ∆vdc_ref.

The voltage adjustment by the virtual inertia controller

should be proportional to the change of the grid angular

frequency ∆ωr in order to generate an appropriate amount of

virtual inertia, while the voltage/current controller simply

regulates the DC-link voltage vdc to follow the voltage

reference adjusted by the virtual inertia controller.

Detailed control schemes will be elaborated in the

following sections based on the system and control

parameters in Table I. It should be highlighted that the

adopted virtual inertia control regulates the DC-link voltage

in proportional to the grid frequency for inertia emulation.

Therefore, it only needs the frequency signal rather than the

df / dt signal, and hence it is different from the df / dt-based

scheme. In addition, the adopted virtual inertia control is

applied to “grid-following” power converters, which are

controlled as AC current sources and cannot operate alone in

the islanded mode.

III. INSTABILITY UNDER WEAK GRID CONDITIONS

This section will detail the control philosophy and explore

the instability for the three-phase grid-connected power

converters with the virtual inertia control under weak grid

conditions.

A. Control without Virtual Inertia

The relationship between the converter voltages vcabc and

grid voltages vabc can mathematically be described by the

following equation in the synchronous dq-frame [39]:

0

0

( )( ) ( ) ( )

( ),

( ) ( ) ( )

cd

cd d t t cq

cq

cq q t t cd

tt t t

tt

div v L L i

dt

div v L L i

dtt t

= + −

= + +

(4)

where vcd(t) and vcq(t) denote the d- and q-axis components of

converter voltages, vd(t) and vq(t) represent the d- and q-axis

components of grid voltages, icd(t) and icq(t) designate the d-

and q-axis components of converter currents, respectively, Lt

is the total inductance (i.e., Lt = Lc + Lg), and ω0 stands for

the fundamental angular frequency. The terms −ω0Lticq(t) and

ω0Lticd(t) are caused by the cross-coupling effect between the

Fig. 1. Inertia mapping between the synchronous generator and the capacitor [29].

Fig. 2. Schematic diagram of a three-phase grid-connected power

converter with the virtual inertia control (PWM stands for pulse-width

modulation).



d- and q-axis. Although the cross-coupling effect due to the

filter inductance Lc may be suppressed by the current

feedforward control, the effect of the grid inductance Lg can

be dominant in a weak grid condition, and this effect is

difficult to be predicted and eliminated [1]. To facilitate the

analysis, the current feedforward control is not included here.

The system in (4) can be expressed in the complex frequency

domain as

0 plant

0 plant

( ) ( ) ( ) ( )

(

( ),

() ( ) ( ) ) ( )

cd d t cq cd

cq q cd cqt

v v L i G s is s s s

s s L sv v i s i sG

+ =

−

− − =

(5)

where Gplant(s) can be represented by

plant

1( ) .

t

G sL s

= (6)

Proportional integral (PI) controllers are employed as the

current controller Gi(s) and voltage controller Gv(s),

expressed as

( ) ,ci

i cp

KG s K

s= + (7)

( ) ( ),vi

v vp

KG s K

s= − + (8)

where the minus sign is caused by the definition of converter

current directions in Fig. 2. Considering the time-delay

introduced by reference computations and pulse updates,

whose effect can be simplified as a first-order lag to

approximately model its low-frequency characteristics for

simplicity [38]:

1( ) ,

1d

d

G sT s

=+

(9)

in which Td = 1.5 / fs with fs being the sampling frequency,

the control architecture of the cascaded voltage/current

controller is shown in Fig. 3. It is clear from Fig. 3(a) that the

DC-link voltage vdc is regulated to its reference vdc_ref

(normally it is a constant Vdc_ref) through the control of the d-

axis current icd. Variations in the d-axis current icd will

directly affect vdc through a transfer function Giv(s), which is

caused by the real-time power balance between the AC-side

and DC-side of the three-phase power converter. Under the

assumption of an ideal lossless conversion, it can be derived

as

dc_ref dc

3( ) ,

2

d

iv

VG s

V C s

−= (10)

where Vd denotes the rated value of vd and Vdc_ref stands for

the reference value of vdc. The grid synchronization can be

achieved using a PLL. It enables the detection of the phase-

angle from the PCC voltages vgabc. According to [38] and

[39], the PLL will influence both the d- and q-axis current

control due to the transformations between natural frame and

synchronous frame. When the grid-connected power

converter is operated with a unity power factor, the influence

of the PLL on the converter control is reflected in the q-axis.

As shown in Fig. 3(b), the extra terms Icd∆θpll and Vd∆θpll are

introduced by the PLL. Icd denotes the rated value of icd, and

∆θpll represents the difference between the phase-angle

locked by the PLL and that of PCC voltages. Although the

Table I. System and control parameter values.

Description System parameter Description Control parameter

Symbol Value Symbol Value

DC-link voltage reference

Vdc_ref 400 V PLL proportional gain Kpll_p 3 (rad/s)/V

Filter inductance Lc 2 mH PLL integral gain Kpll_i 300 (rad/s)/(V∙s)

Grid inductance Lg 5 mH Current proportional gain Kcp 15 V/A

DC-link capacitance Cdc 2.82 mF Current integral gain Kci 300 V/(A∙s)

Grid voltage amplitude Vd 155 V Voltage proportional gain Kvp 0.2 A/V

Power rating Sbase 1 kVA

Voltage integral gain Kvi 2 A/(V∙s)

Sampling/switching frequency fs / fsw 10 kHz Inertia control gain Kωv / Kωv_pu 14.32 / 11.25 V/(rad/s)

Frequency reference f0 50 Hz Time delay constant Td 1.5 / fs

Generator inertia constant H 5 s Virtual inertia coefficient HcapKωv_pu 2.5 s

(a) Control in d-axis (b) Control in q-axis

Fig. 3. Voltage/current controller implemented in the synchronous reference frame.



PLL is assumed to only affect the q-axis current control, it

should be noted that the d-axis current icd will influence the

PLL (e.g., its phase angle ∆θpll) through the coupling between

d-axis and q-axis, and this effect will be analyzed in the

following sections.

A small-signal model of the PLL is given in Fig. 4, from

which the relationship between ∆θpll and vgq can be derived

and represented as [38]

pll pll_ pll_

pll 2

pll_ pll_

( )( ) .

( )

p i

gq d p d i

s K s KG s

v s s V K s V K

+= =

+ + (11)

where Kpll_p and Kpll_i denote the proportional and integral

gains of the PLL loop filter (i.e., a PI controller), respectively.

Detailed derivations of the PLL model and its influence on

the current control are elaborated in [38] and [39]. It is

revealed that the PLL proportional gain Kp rather than the

PLL integral gain Ki will play a dominant role in determining

the system stability [39]. An excessively large Kp will

destabilize the power converter control under weak grids. In

contrast, an excessively small Kp will deteriorate the PLL

dynamics. In this paper, Kpll_p and Kpll_i are designed to yield

a crossover frequency of 75 Hz and a phase margin of 75˚ for

the PLL control. This design guarantees the stable operation

of power converters without the virtual inertia control as well

as fair PLL dynamics.

The closed-loop transfer functions of the d- and q-axis

current control Gid_cl(s) and Giq_cl(s) can be derived from Fig.

3 as

plant

_cl

_ref plant

( ) ( ) ( )( )( ) ,

( ) 1 ( ) ( ) ( )

i dcd

id

cd i d

G s G s G si sG s

i s G s G s G s= =

+ (12)

_ cl

_ref

plant

pll plant

( )( )

( )

( ) ( ) ( ).

( ) ( ) ( ) ( ) ( ) ( )

cq

iq

cq

t i d

t g d cd i d t i d

i sG s

i s

L G s G s G s

L L G s G s I G s V L G s G s G s

= =

− + +

(13)

Furthermore, the voltage-loop gain without the virtual inertia

control Gv_ol_wo(s) is derived as

dc

_ ol_wo _cl

dc

( )( ) ( ) ( ) ( ).

( )v v id iv

v sG s G s G s G s

v s= =

(14)

Moreover, the closed-loop transfer function of the voltage

control without the virtual inertia control Gv_cl_wo(s) can be

represented as a function of Gv_ol_wo(s) as

_ ol_wodc

_ cl_wo

dc_ref _ ol_wo

( )( )( ) .

( ) 1 ( )

v

v

v

G sv sG s

v s G s= =

+ (15)

Fig. 5 illustrates the Bode diagram of the voltage-loop

gain without the virtual inertia control Gv_ol_wo(s), where a

positive gain margin of 44 dB can readily be obtained,

indicating a large stability margin. Fig. 6 illustrates the pole-

zero map of the closed-loop transfer function of the voltage

control without the virtual inertia control Gv_cl_wo(s), and the

corresponding zoom-in inset is also provided. Since all the

closed-loop poles are in the left-half-plane, the voltage and

current control under weak grid conditions can always be

stable with the parameter values listed in Table I.

B. Control with Virtual Inertia

Referring back to Fig. 3(b), because of the coupling effect

between the d- and q-axis, variations of icd will change ∆θpll.

Considering the coupling effect, it is possible to derive the

transfer function from icd to ∆θpll, i.e., Gid_∆θ(s), as

pll

_ 0 _ cl pll

( )( ) ( ) ( ).

( )id g iq

cd

sG s L G s G s

i s

= = (16)

The Bode diagram of Gid_∆θ(s) is illustrated in Fig. 7. It is

obvious that Gid_∆θ(s) exhibits a low-pass filter characteristic.

As compared to that of icd, the magnitude of ∆θpll has been

Fig. 6. Pole-zero map of the closed-loop transfer function of the voltage

control without the virtual inertia control Gv_cl_wo(s).

Fig. 4. A linear small-signal model of the PLL.

Fig. 5. Bode diagram of the voltage-loop gain without the virtual inertia

control Gv_ol_wo(s).



attenuated more than −40 dB. This is because the PLL

transfer function Gpll(s) can be approximated to a gain (1 / Vd)

in the low frequency band according to (11), and Vd is greater

than 100 V here. As a result, the d-axis current control has

almost no effect on the PLL, and thus the coupling effect

between the two axes are often ignored when analysing the

instability issue due to PLLs for simplicity [39].

By directly linking the change of the capacitor voltage

reference ∆vdc_ref and the change of the grid angular frequency

∆ωr through a proportional gain Kωv, virtual inertia can be

provided by the three-phase grid-connected converter [29]. It

should be emphasized that ∆ωr is normally obtained from the

PLL in practice. Mathematically, ∆ωr equals the time-

derivative of the phase-angle ∆θpll, i.e., ∆ωr = ∆θplls, as can

be observed from Fig. 4. Therefore, the virtual inertia control

inherently introduces a differential operator between ∆θpll

and ∆vdc_ref, as shown in Fig. 8, where the entire control

structure for the three-phase power converter with the virtual

inertia control is given. Note that ∆ωr is expressed as the time

derivative of ∆θpll in Fig. 8, while ∆θpll is represented as the

integral of ∆ωr in Fig. 4. In fact, these two representations are

equivalent.

According to Fig. 8, the transfer function from icd to

∆vdc_ref, denoted as Gid_∆vref(s), is given by

dc_ref

_ ref _

( )( ) ( ) .

( )id v id v

cd

v sG s G s K s

i s

= = (17)

Although the differential operator in (17) inserted between

∆ωr and ∆θpll is quite necessary for inertia emulation, it also

makes ∆vdc_ref very sensitive to icd, particularly for a large

value of Kωv, as evidenced by the magnitude diagram of

Gid_∆vref(s) shown in Fig. 9, which serves to illustrate the

effect of the virtual inertia control. The mechanism for

magnitude amplification can be explained by the transfer

function Gpll_ωr(s), namely ∆ωr(s) / vgq(s), which is essentially

the multiplication of the PLL transfer function in (11) and the

differential operator s:

2

pll_ pll_

pll_ 2

pll_ pll_

( ) .p i

r

d p d i

K s K sG s

s V K s V K

+=

+ + (18)

As compared to (11), the differential operator reshapes

the PLL transfer function from a low-pass filter Gpll(s) to a

high-pass filter Gpll_ωr(s). In addition, Gpll_ωr(s) can be

approximated as a gain of Kpll_p in the high-frequency band.

Since Kpll_p > 1 (rad/s)/V, the magnitude will be amplified by

the differential operator. The effect of magnitude

amplification, together with the considerable phase-shift of

Gid_∆vref(s), tends to destabilize the power conversion system,

and the instability issue will be disclosed as follows.

It can be obtained from Fig. 8 that the branch from icd to

∆vdc_ref is in parallel with the branch from icd to vdc. First, note

that one branch links icd to vdc through Giv(s). Additionally,

the other branch starts from the term icdω0Lt (i.e., icd

multiplied by ω0Lt), goes through the q-axis current control

Fig. 7. Bode diagram of Gid_∆θ(s), i.e., the transfer function from icd to

∆θpll.

Fig. 8. Entire control structure for the three-phase power converter with the

virtual inertia control.

Fig. 9. Bode diagram of Gid_∆vref(s), i.e., the transfer function from icd to ∆vdc_ref with the conventional virtual inertia control.



to ∆θpll, and finally reaches ∆vdc_ref through the multiplication

of Kωv and s. Since the inputs of the two branches (namely

icd) are the same, and the corresponding outputs are operated

in the same summing node, the two branches are in parallel.

The additional branch from icd to ∆vdc_ref essentially models

the effect of the virtual inertia control, which may cause the

instability issue, as will be detailed. Substitution of (16) into

(17), it is noted that the gain of this additional branch is

determined by the grid inductance Lg. If Lg = 0 mH, the

additional branch will be disabled. Therefore, the instability

issue will not occur under stiff grids where Lg = 0 mH. In

addition, the increase of Lg will intensify the influence of the

virtual inertia control on the system stability. Since the two

branches are in parallel, they can be combined with ∆vdc_ref

being regarded as a part of vdc. In this sense, (vdc − ∆vdc_ref)

will be the output of the modified voltage-loop in

replacement of vdc. The loop-gain of the modified voltage-

loop with the conventional virtual inertia control Gv_ol_wc(s)

can be derived as

dc dc_ref

_ ol_wc

dc

_cl _ ref

( ) ( )( )

( )

( ) ( ) ( ) ( ) .

v

v id iv id v

v s v sG s

v s

G s G s G s G s

−=

= −

(19)

The Bode diagram of Gv_ol_wc(s) is drawn in Fig. 10, where

the negative phase margins and gain margins clearly

demonstrate the instability of the voltage control. Moreover,

the similarities between the magnitude curves of Fig. 9 and

Fig. 10 indicate that Gid_∆vref(s) may greatly influence the

voltage control because of its magnitude amplification effect.

To further verify the instability of the voltage control, the

closed-loop transfer function of the voltage control with the

conventional virtual inertia control Gv_cl_wc(s) is derived as

dc

_ cl_wc

dc_ref

_cl

_cl _ ref _cl

( )( )

( )

( ) ( ) ( ).

1 ( ) ( ) ( ) ( ) ( ) ( )

v

v id iv

v id id v v id iv

v sG s

v s

G s G s G s

G s G s G s G s G s G s

=

=− +

(20)

Substituting of (14) into (15) and then comparing (15) with

(20), it is noted that an additional term

−Gv(s)Gid_cl(s)Gid_∆vref(s) is added to the denominator of

Gv_cl_wc(s), and this term causes the instability of the voltage

control.

Fig. 11 presents the pole-zero map of the voltage control

with the conventional virtual inertia control, where a gain Kωv

of 14.32 is used and henceforth, as given in Table I. As

observed, a pair of conjugate poles appearing in the right-half

plane (RHP) will make the voltage control unstable, which is

consistent with Fig. 10. Although the presented stability

analysis is on the basis of a single closed-loop transfer

function of the voltage control, it essentially describes the

relationship between the most concerned input and output of

practical multiple input multiple output power conversion

systems.

IV. PROPOSED MODIFIED VIRTUAL INERTIA CONTROL

FOR STABILITY ENHANCEMENT

According to the previous analysis, the magnitude

amplification of the differential operator should be

responsible for the system instability. In this section, the

differential operator is intentionally modified in order to

reduce its high-frequency amplification effect, and thus to

improve the system stability.

As mentioned before, Gpll_ωr(s) can be simplified into

Kpll_p in the high-frequency band. This is because of the

second-order term (Kpll_ps2) in the numerator of Gpll_ωr(s) in

(18). The proposed control scheme aims to reduce the

coefficient of this second-order term. Its fundamental

principle is shown in Fig. 12, where the change of the

modified angular frequency ∆ωm will be used instead of ∆ωr

to generate ∆vdc_ref through the proportional gain Kωv. The

transfer function Gpll_ωm(s) from vgq to ∆ωm can be derived

from Fig. 12(a) as 2

pll_ pll_

pll_ 2

pll_ pll_

( )( ) ,

p m i

m

d p d i

K K s K sG s

s V K s V K

− +=

+ + (21)

Fig. 10. Bode diagram of the voltage-loop gain with the conventional

virtual inertia control Gv_ol_wc(s).


control with the conventional virtual inertia control Gv_cl_wc(s).



where Km denotes the proportional gain of the proposed

modified virtual inertia control. Notice that this control

scheme will not affect the frequency detection when the PLL

locks the grid voltage, since vgq_pll = 0 in steady state. By

comparing Gpll_ωr(s) in (18) and Gpll_ωm(s), it is noted that the

proposed control scheme essentially attaches the following

transfer function to the differential operator:

pll_ pll_ pll_

pll_ pll_ pll_

( ) ( )( ) ,

( )

m p m i

m

r p i

G s K K s KG s

G s K s K

− += =

+ (22)

where Gm(s) would be a lead-lag compensator when Km ≠

Kpll_p or a low-pass filter if Km = Kpll_p. Considering (22), the

transfer function from icd to ∆vdc_ref should be reorganized as

Gid_∆vref_wp(s):

dc_ref

_ ref _ wp _

( )( ) ( ) ( ) .

( )id v id m v

cd

v sG s G s G s K s

i s

= = (23)

The Bode diagram of Gid_∆vref_wp(s) is plotted in Fig. 13. As

Km increases and approaches Kpll_p (Km ≠ Kpll_p), the

magnitude of Gid_∆vref_wp(s) drops without any compromise on

the phase-shift, particularly in the high-frequency band which

is more of concern. When Km reaches Kpll_p (Km = Kpll_p), the

dramatic attenuation of the magnitude of Gid_∆vref_wp(s) can be

achieved at the expense of an additional 90-degree phase lag.

Referring to (22), this additional phase lag is caused by the

degradation of the numerator order of Gm(s) when Km = Kpll_p.

Regardless of phase-shifts, the proposed control scheme

manages to attenuate the magnitude amplification, and this

result can be beneficial in terms of system stability.

Furthermore, the loop-gain of the modified voltage-loop

with the proposed virtual inertia control Gv_ol_wp(s) is written

as

dc dc_ref

_ ol_wp

dc

_cl _ ref _ wp

( ) ( )( )

( )

( ) ( ) ( ) ( ) .

v

v id iv id v

v s v sG s

v s

G s G s G s G s

−=

= −

(24)

Fig. 14 illustrates the Bode diagram of the voltage-loop

gain with the proposed virtual inertia control Gv_ol_wp(s),

where the stability enhancement can clearly be observed.

When Km = 0.5Kpll_p, although the system featuring a negative

phase margin and a negative gain margin is unstable, both

margins are improved. The case of Km = Kpll_p results in a

system with a gain margin of 4.35 dB and a phase margin of

34.1 degrees, and these positive stability indices indicate the

system is stable.

The closed-loop transfer function of the voltage control

with the proposed virtual inertia control Gv_cl_wp(s) is derived

as

(a) Local control structure

(b) Overall control structure

Fig. 12. Fundamental principle of the proposed control scheme.

Fig. 13. Bode diagram of Gid_∆vref_wp(s), i.e., the transfer function from icd

to ∆vdc_ref with the proposed virtual inertia control.

Fig. 14. Bode diagram of the voltage-loop gain with the proposed virtual

inertia control Gv_ol_wp(s).



dc

_ cl_wp

dc_ref

_cl

_cl _ ref _ wp _cl

( )( )

( )

( ) ( ) ( ).

1 ( ) ( ) ( ) ( ) ( ) ( )

v

v id iv

v id id v v id iv

v sG s

v s

G s G s G s

G s G s G s G s G s G s

=

=− +

(25)

The Pole-zero map of Gv_cl_wp(s) is further provided in

Fig. 15. Note that all the closed-loop poles of the voltage

control are located in the left-half-plane, thereby

demonstrating a stable system. It should be commented that

as long as the pole of Gm(s) locates leftwards to its zero, i.e.,

|Kpll_p − Km| < Kpll_p, the proposed control scheme will enable

magnitude attenuation and consequently, the stability is

improved.

V. EXPERIMENTAL VERIFICATIONS

The instability issue of the three-phase grid-connected

converters with the virtual inertia control will further be

experimentally investigated in this section. Moreover, the

effectiveness of the proposed modified virtual inertia control

in addressing the instability issue will also be verified. In the

experiments, the grid voltages vabc were formed by a virtual

synchronous generator (VSG), which is essentially a three-

phase power converter exhibiting the same electrical terminal

characteristics as conventional synchronous generators,

providing the base power system inertia and forming the grid

voltages [40]. The reason for the employment of the VSG lies

in the replacement of conventional synchronous generators

and proper regulation of the grid frequency so as to

demonstrate the benefits of virtual inertia. A step-by-step

design of the VSG control, together with its design

parameters, can be found in [29] and [41]. In addition to the

VSG, other parts of the system with the parameters listed in

Table I are shown schematically in Fig. 2. Control systems

were implemented in a dSPACE control platform

(Microlabbox). An eight-channel oscilloscope (TELEDYNE

LECROY: HDO8038) was used to capture all the

experimental waveforms.

Fig. 16 presents the steady-state experimental results of

the three-phase power converter with the conventional virtual

inertia control, where ∆fr (∆ωr = 2π∆fr) denotes the grid

frequency change measured by the PLL, and the other

notations can be found in Fig. 2. In this case, the VSG is

employed solely to provide the grid voltages with a fixed

frequency as an AC voltage source. As observed, the current

waveforms icabc are seriously distorted. It should be

mentioned that the saturation units are necessary in practice

to limit the variation ranges of the DC-link voltage vdc and

frequency fr to avoid over voltages. Due to the saturation

units, vdc and fr can only oscillate within certain ranges rather

than being completely unstable. Therefore, the oscillations in

the grid frequency change ∆fr imply that the virtual inertia

control through the direct link between fr and vdc should be

responsible for the instability. Fortunately, the instability

issue can successfully be addressed once the proposed

modified virtual inertia control is enabled, as demonstrated in

Fig. 17. It is clear that the oscillations in the grid frequency

change ∆fr and the DC-link voltage vdc disappear. Due to the

stable DC-link voltage vdc, the converter currents become

almost distortion-free with negligible variations (induced by

power converter losses). These experimental results agree

well with the pole-zero maps shown in Fig. 11 and Fig. 15. It

should be noted that the instability due to the virtual inertia

control remains even if the PLL is replaced by other more


control with the proposed virtual inertia control Gv_cl_wp(s).

Fig. 16. Steady-state experimental results of the power converter with the conventional virtual inertia control (vabc: the grid voltages, icabc: the

converter currents, vdc: the DC-link voltage, and ∆fr: the frequency change).

Fig. 17. Steady-state experimental results of the power converter with the proposed virtual inertia control (vabc: the grid voltages, icabc: the converter

currents, vdc: the DC-link voltage, and ∆fr: the frequency change).

C2

C4

vabc : [50 V / div]icabc : [5 A / div] vdc : [200 V / div]

fr : [5 Hz / div] Time : [20 ms / div]

C2

C4

C2

C4

vabc : [50 V / div]icabc : [5 A / div] vdc : [200 V / div]

fr : [5 Hz / div] Time : [20 ms / div]



robust synchronization units, such as the frequency-locked-

loop (FLL) [21], or with modified PLL designs.

Through the proposed virtual inertia control, the three-

phase grid-connected power converter may effectively

contribute virtual inertia to weak power grids. For

illustration, Fig. 18 shows the grid frequency and the DC-link

voltage responses of the power converters with and without

the virtual inertia control when they experienced a 5% step-

up load change. Such a step-up load change causes a

relatively serious frequency event, which is sometimes used

to emulate generation or load tripping in real power systems

[5], [42]. It should be mentioned that the step-up load change

causes the demanded power to be greater than the generated

power. As a consequence, the grid frequency drops, which is

seen by all the synchronous generators as a common signal to

increase their respective power outputs for balancing the

power mismatch between generation and demand. In the case

of such a frequency event, it is important to keep the

frequency drop, as well as its changing rate, i.e., the rate of

change of frequency (RoCoF), below the limits defined by

grid codes [3]. Otherwise, excessive frequency drops may

further cause generation and load tripping, thus leading to

cascaded failures. In extreme cases, system blackouts may

even occur [4].

In Fig. 18, the VSG is operated to regulate the grid

frequency in a similar way as conventional synchronous

generators do. As compared to the grid-connected power

converter without the virtual inertia control, the power

converter with the virtual inertia control registers a change of

its DC-link voltage that is proportional to the grid frequency

for inertia emulation during the frequency event. As a result,

the inertia constant is improved from 5 s to 7.5 s, indicating

a 50% enhancement of the power system inertia. The

increased inertia can contribute to an 11% increment of the

frequency nadir (i.e., the lowest frequency point) and a 33%

improvement of the RoCoF, as demonstrated in Fig. 18. Note

that the extent of performance improvements depends on the

emulated virtual inertia, which is further determined by the

DC-link voltage, voltage variation range, and DC-link

capacitance, as detailed in [29].

VI. CONCLUSION

This paper has identified the potential instability issue for

three-phase grid-connected power converters with the virtual

inertia control when they are connected to weak power grids.

The exploration indicates that the virtual inertia control

introduces an extra link between the d-axis current and the

DC-link voltage reference through the d- and q-axis coupling,

q-axis current control, phase-locked-loop, and virtual inertia

controller. Specifically, there is an inherent differential

operator inside the link, which is responsible for the

instability, when the converter is connected to weak grids.

Accordingly, the virtual inertia control has been intentionally

modified in order to mitigate the amplification effect

introduced by the differential operator. As a result, three-

phase grid-connected power converters can effectively

contribute virtual inertia to weak power grids, which have

been verified by the experimental results. The major

contributions of this paper can be summarized as follows:

(1) Identify the instability issue faced by three-phase grid-

connected power converters with the virtual inertia control

under weak grids;

(2) Explain the mechanism for the instability issue;

(3) Propose a modified virtual inertia control scheme to

resolve the instability issue.

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Jingyang Fang (S’15) received the B.Sc. and M.Sc. degrees in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 2013 and 2015, respectively, and the Ph.D. degree from the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, in 2018.

From May 2018 to August 2018, he was a Visiting Scholar with the Institute of Energy Technology, Aalborg University, Aalborg,

Denmark. He is currently a Research Fellow with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He has published more than 40 papers in the fields of power electronics and its applications, including 16 top-tier journal papers. His research interests include power quality control, stability analysis and improvement,



renewable energy integration, and digital control in more-electronics power systems.

Dr. Fang was a recipient of the Best Paper Award of Asia Conference on Energy, Power and Transportation Electrification (ACEPT) in 2017 and a recipient of the Best presenter of 2018 IEEE International Power Electronics and Application Conference and Exposition (PEAC) in 2018.

Pengfeng Lin (S’16) received the B.Sc. and M.Sc. degree in electrical engineering from Southwest Jiaotong University, China, in 2013 and 2015 respectively. He is currently working toward Ph.D degree in Interdisciplinary Graduate School, Eri@n, Nanyang Technological University, Singapore.

His research interests include energy storage systems, hybrid AC/DC microgrids and electrical power system stability analyses.

Hongchang Li (S’12−M’16) received the B.Eng. and D.Eng. degrees in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 2011 and 2016, respectively.

From August 2014 to August 2015, he was a Visiting Scholar with the Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA, U.S. He is currently a Research Fellow with the Energy Research Institute at Nanyang Technological University, Singapore.

His research interests include wireless power transfer, electron tomography and distributed energy storage systems.

Yongheng Yang (S’12-M’15-SM’17) received the B.Eng. degree in electrical engineering and automation from Northwestern Polytechnical University, Shaanxi, China, in 2009 and the Ph.D. degree in electrical engineering from Aalborg University, Aalborg, Denmark, in 2014.

He was a postgraduate student at Southeast University, China, from 2009 to 2011. In 2013, he spent three months as a Visiting Scholar at Texas A&M University, USA. Dr. Yang is currently an

Associate Professor with the Department of Energy Technology, Aalborg University. His research focuses on the grid integration of renewable energy, in particular, photovoltaic, power converter design, analysis and control, and reliability in power electronics. He has contributed to two books on the control of power converters and grid-connected photovoltaic systems.

Dr. Yang is an Associate Editor of the IEEE JOURNAL OF EMERGING AND SELECTED TOPICS IN POWER ELECTRONICS (JESTPE), the CPSS Transactions on Power Electronics and Applications, and the Electronics Letters. He was the recipient of the 2018 IET Renewable Power Generation Premium Award.

Yi Tang (S’10−M’14−SM’18) received the B.Eng. degree in electrical engineering from Wuhan University, Wuhan, China, in 2007 and the M.Sc. and Ph.D. degrees from the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, in 2008 and 2011, respectively.

From 2011 to 2013, he was a Senior Application Engineer with Infineon Technologies Asia Pacific, Singapore. From 2013 to 2015, he was a

Postdoctoral Research Fellow with Aalborg University, Aalborg, Denmark. Since March 2015, he has been with Nanyang Technological University, Singapore as an Assistant Professor. He is the Cluster Director of the Advanced Power Electronics Research Program at the Energy Research Institute, Nanyang Technological University.

Dr. Tang was a recipient of the Infineon Top Inventor Award in 2012, the Early Career Teaching Excellence Award in 2017, and four IEEE Prize Paper Awards. He serves as an Associate Editor for the IEEE Transactions on Power Electronics (TPEL) and the IEEE Journal of Emerging and Selected Topics in Power Electronics (JESTPE).

an improved virtual inertia control for three-phase

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